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Full text of "The Calculation of Compressible Flows with Local Regions of Supersonic Velocity"

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1114 THE CALCULATION OF COMPRESSIBLE FLOWS WITH LOCAL REGIONS OF SUPERSONIC VELOCITY By B. Gothert and K. H. Kawalki Translation Berechnimg kompressibler Stromungen mit brtlichen tjberschallfeldern Forsch-ungsbericht Nr. 1794 '^glE^ 1 - Washington I \ March 1947 3 1176 014412259 "SAnmasiJiL adtjecuri caisxsnuEE for jqsrgkautics TBCHErXCAL MEMQBAinXJM NO. llllv TEE CALCDLATIOCT 093* CCMPSSSSIELS FLOWS VIXH LOCAL HBGrXORS 0? SUFEBSQinJC vsLocrn^ By B. Gothart and. K, E. EamUd ABSTRACT: Xbe follovlng report lo ooxLcenxod vLth a metbod for the approxlmato oalculatlon of comproasl'ble flows abont pznflloB vlth locel regions of supersonic velocity. Tlw flow oi'ounfl. a slender profile Is treated as oil exampll.e. OIMJHE! I. Startenent of the PrpWLam. H. Stirvey of the I^i^thod used. III. Colculatloaa of the Bieurrple. 17. Appro3dinate Treaiaiont of Local BpglonB of Supernonlo Telocity. V, Synanoti'loal and Unarnsnetrlcal Koslone of Buper sonic Flow. VI. Summary. I. STATSIEWP 0? THE PROBEE-I Sevei<al methods ore Tmovn tor the calculation of ccmpresslhle flovs at hl£^ subBonlc velocities. The resulting approximate BolutlonB are quite -osefol as long aa sound velocity is not exoeeded at any point of the flov field. Hovever, apparently all these approximate cELLoulations, t/lthout exception, cease to converge or to render useful flov pattozna If the condition of purely suhsonlc flov Is no longer satisfied. Moreover, numerous tests In vlnd tunnels conf limed the result that the reconversion *"Berechnung JcdnpresBlhler StrflBrunson mit b'rtllchen Uber- echallfeldem." Zentrale f(£r vlssenechaftllches Berlchtswesen der Luftfahrtforschung des Generalluftzeugmslsters {2XIB) Berlln- Adlerahof , Forechungaherlcht JTr. 179'^> Berlln-Adlershof, den 7. August 19'+3. DACA IK So. llll^ of local supareoQlo flovB Into subsonic flovs la real flovs takSB place not steeuUlT- Imt generally "by means of a oocoipreSBlon slioclc which completely cbenf^s the vdiole flov pattezn^^ There- fore, the queetion eriees Aether a contlnuouB process from supersonic to subsoolo flov Is at all possllxLe for "bodies In parallel flow, even thoue^i the foxmatlon of a "bovasdory layer on the surface of the "body Is at first neQlected. An appyoxlBifltlon me-Qiod vas tested In the JfTL In order to olaarlfy these questions; this nusthod maJces possible the calculation of flows with local regions of supersonic velocity. II. SURVEY OV TBS MBraor TEED The devolopnent of the method started frcm the fact' that the Vnovn approximation msthodo for ccmpreselhls flows, tdilch are without oiooptlon "based oa a step-by- step Improvecent of tlie Inccoipresslble flov, are quite appropriate for tho llTolted liomaln of the pure suboonlo flov eztendlnf; Srcsa. iJiflnlty to the sonic voloolty boundary In the flov field near tiie body. Eovever, for the region of the local supersonic ilov a method based on the pxt^pertles of s'lpersonlc flov will be subsequently used, as for Instonco the method of characteristics of Prandtl-Busemazin. Accordingly the partial areas of supersonic and subsonic flow, respectively, are calculr.ted SAparately by different methods \dilch In each case arc treated according to the peculiarities of the partial flow to be calculated. The partial flows that wore thus detezmlned must then 'bo Joined in such a manner that the flows agree on the surface of contact of the two regions, that is, on tho sonic volocitj" boundary, with respect to magnitude as well as to direction of the velocity. Figure 1 shove a scheoamtic representation of the boundary between the two flov areas. 1. Subsonic Region with analT Perturbation 7eloolties Frandtl's rule will repi-esent a good approximation for a great part of the outer subsonic flov area, eijooe the perturbation velocities caused by the profile are sufficiently small up to Bcaae distance from the sonic boundary so that the assunrptiona of no account will be taken here of flows -idilch ali^tlj' exceed sound velocity because for them there is no certain way of distingui^lng between a steady trsnsJtlon and a cccrpresslon shock; neither by pressure -distribution measureoaents nor by optical observations, for Instance according to the schlleran method. . KAGA mso, mk FraDcLtl's approximation are rather veil aatlsfled. The Bubsoolc flov of this regloa-can, of cotErse, not "be o'btalned "by Blnsply dlstorttog the Inoampresslhle flov around the profile aocordlng to PrQiadtl*B rule. First, the local siiper sonic velocity field requires essentially more spaoe than the suhsonlc flov. The outer subsonic streamlines, therefore, ore vldened outvazrL not only "by the hody In the flov tut also hy the additional displace- ment due to the super sonla reloclty field. "Eoverer, this additional vldenlng of the streazallnoa ly the aupersonlo Telocity field can only affect the subsonic flov like a modification of the Isoundary cooodltlons at the sonic Telocity houndary; such a modification may he represented In a single way hy source end sink distrlhutlons, dlpoles, end so forth. 2. Suhsonlo Eoglon vith El^ Porturhrtion Telocltles As mantloDod "before, the pMrt of the siibsonlc flow vfaich has t^ Tue calculated according to Pi'andtl does net extend as foi^ as the sonic houndary; It only reachos up to a hounlrtry lino near the sonic houndory determined "by an acreod siifflclontly small Talue of the perturhatlcn Tol^oltles. (Tlila houndorj'- line is represented In fig. 1 hy a datfied lino.) TIio calciilation for the rejjlon trcm this line to the sonic Telocity 'bouadary must generally "be csLLTlod out hy an ImproTed suhscnic method. A nunorlcal method of calculation seems to he particularly appro- prtate for this Inteimndiate region. The oiitor flow according to Prandtl, which io assumed as known, nujy "by tiio mothod hy continued a little further, namely up to the sonic Telocity "boundary. One assumes, for Instance, that the flow field is coTered "by a rectangular grid of selected points. The Telocity cor^ononts T^: and Ty at these points in the outer field are knotm for each case according to. Prandtl* s approximation. Then the Telocltles at the inner grid points may he calculated fixim the Imown Taluss at the outer grid points if the differential quotients vlilch are declslTe for this continuation are appro- ximated "by the corresponding difference quotients.^ Figure 2 shall "be considered as an ex&mple. Magnitude and direction of the Telocltles at all grid points outside of the "boundeury lines are assumed as known. The air density shell he known also. Then the exact equations of the continuity and the Irrotationality are, for the indicated grid point "2"! 21 wish to express my thanks here to Dr. H. Schu"bert/D7L for his suggestion to treat ccmpresslhle flc/'s according to the difference method. HAGA m No. llll^ > or, In the notation of difference giiotlentEi! •^1 ^^3 ■ ^^1 Va ■ V ■ ' J B- If one asBranes the grid to "be qijadratlc vlth neahes of equal vldth then ^ a ZSy and the equatlone gl7en aliove are eliapllf led as follows: 'X2 ^ ^A,k * P2 ^ ^^r.A,k ■ -V2 ^ ^^,1 - P2 ^ ^^73,1 (1*) "^^A,!. - ^^X3,i (It) Frcm the second eqiiatlon one Immediately obtained the Telocity component y-.. at the point A, -while the first equation gives a relation hetvean the velocity ccairponont v and the air density at the point A. Velocity and olr density are connected In a rather ccanplicated way hy the adlahatlc relation Pc -^H'<@f (p heire represents the eJLr density In the gas at rest a the critical Bo-imd velocity) . ITherefore, It will he praotloal to use the linearized relation Ap ■ j^ Av instead of the esaot ftmctlon, the error "beinc nogllgihle, heoanse of the snuQl width of the meshes: HACA TM So. mh l(p/Po) _-2,^. /pNi — — — X --^ X + 1 ^ -^(#*-&) Ccmpare figure 3. With dT n -* X dT_ + -*-x dv" there results T It ■ y ra. and "by siibatltutlon ^(p/Pq) X At-. , + -Z£.x ^, 3CA,3t ^A,kj At. Va-^ ^Ajlt (^) . T ▼ "~: O + D ■ ■■ ■"— d(p/pj d^v/i (3) The Telocltj coiriponentB t a-^jo ani T.^. for the new grid pclnt ■ i/A ero known, according to those ccLLciJ-atlona, frcm ecLne.tlone (1 "b) and (3). Therefore- the air density also Is Imown accordlnf! to eqitatlon (2). Thus the Intezsiedlate region between the- subsonic flow with nr.wn perturlsatlon Telocltlos (calculation accordlnii to Premltl) and the 'boundaxy may Tae detexmlned end the position cf the sonic Telocity Ixjimdory and the Telocity direction alone that "boundary will Tie o^btalnod as final result. It will haTe to "bo further Investigated whether a calculation of the wliole s-jper sonic velocity field Tieyond the eonlc houndary up to the surface of the Txidy "by this method would "be practical. I!a the main, the time spent on the calculation H and the accuracy of the method which has "been developed here as contrasted with the metAiod of oliaracterlstlcs of Prandtl-BuBQBiaiin will haTe to "be considered. The difference method ^Aiich has "been cLeTelo?ed here does not, howerer, pexnlt a start of the numerical calculation at an arlaltrazy dlstanoe frcm the liody In the flow; for at a very great distance frcan the "body the calculation would not be euff lolently accurate since the moshes cf the grid v:<uld 1)0 too small. This connluslan can bo d.^avn from the foot tliat at a gireat distance the flow about any body without external forces can be replaced, except for "f"*^'''' ITACA mNo. ink deviations, "bgr the flov field around a single dlpole. It Is precisely tlie small derlatlons from the dlpole flov, lAiich exist even at the greatest distance, that pezmlt a continuation of the flov, for Instance according to the difference method, not a1x>ut a cylinder "but around the special profile. 3. ^tgpersonlo Beglon I^ram the calculations descrllied &Taove, vhlch were carried out In the subsonic area according to Pran'ttl's rule, or hy the difference method, the hoiaidary line of the supersonic velocity field vas obtained. The Telocity along -Uils lino equals the sound Telocity and the dlr-ectlom ef the flov at every point of this line Is Imovn. The next problem consists In Joining the corresponding sitpersonlc velocity field to this boundary line; for Instance, by the method of oharewterl sties of Prandtl- Busemann. According to the method of characteristics it is Imovn that the changes of state occurring In a supersonic flov are manifested In e^cpanslon cr ccnpresslon vaves. Therefore, ■vhen the flov passes throu^ thdtse lines of disturbance, the flov velocity and the flov diretttloii are modified by definite values 'Which can easily be detemlned from the graphical representation of the chai-acteristicsi® It can be Inferred fjxm the condition of constant velocity at the sonic boundaz? \ipjo that tvD vaves must originate at every point of the siExzU'c'^undary, namely an expansion and a compression vave. (See fig, h.) -Pie velocity direction Junips from one starting point of these waves at the sonic boundary to the next by a fixed amount for each case. Since these starting points of the vaves can be shifted in snj vay along the sonic velocity boundary, the beginning of a graphical representation of characteristics may be dravn for any given distribution of direction on the sonic boundary. Lllcevlse the starting points of the vaves may be shifted not only on the sonic boundary but also toirards points outside of the sonic boundary, so that the graphical representation of chai'acteristlcs nay be adjusted to any shape of the sonic boundary line. TherevLth it has been demonstrated that at least in certain cases there exists a possibility of continuing the subsonic flov Into a corresponding supersonic flov; finally there results the nev profile contour of the body in the flov/- ^Compare L. Prandtl: "Fuhrer durch die StromungBlehre . " Verlag Vloveg and Sohn, Braunschvelg (19^2) S. 257 und folgende. *In order to render possible sufficiently accurate dravlngs of grids of characteristics for local supersonic velocity f leldjS, a graphical representation of charactoristics vith an interval of 1/5° has been completed at the STL; copies may be obtained. BACA OMiro. ink in. cALcncjfflioir op ah siamfee for a mixed SDEEaasoNic-saBigoiiic itxw A Blmple exsoDople will demanstrate the coAublnsd effect of the calculation method^ for various flow reeXariB, TOie velocity field for a two-dJmeiislanal Blonder "body vae determined In the proximity of the Ixjdy "by means of comformfll trwn af oimatlon and tlie associated distortion according to Prandtl for the Maoh number M » 0.86.- Thus -Uiere resulted a sonic velocity "boundary as Indicated In. figure 2(a). This sonic 'boundary, iftxloh vas ohtalned according to Prandtl*a method, could "be further litiproved "by continuing the flow step "by step to th^ Improved sonic velocity Twundnry; one would have to start from a "boundary lino with sufficiently antfin perturhatlon velocltios iistng the difference method descrihed ahove. Thio corrfictlon of the sonic velocity "boundary vaa at first diareiarded for the sako of simplicity, Elnco in c©neral, it can "bo noglectod for tho "basic calculatlona pltnned for this report. By means of the directions of velocity alone "^e sonic "boundary, the neb f,f characteristics for the Eiuporsonic flow can "be drawn so that the contour of the "body in the flow la a streami- line of this siiporscnic field. Only expansion waves start from the Bvirface of the "body, t&iile cantpresslon vaves, Tiilch are necessecry for a roconversion of the supersonic flew into a sub- sonic flow start witliout exception from the sonic "boundary. This "behaviour la an escontial criterion of the grid of characteristics. The manner of reconversion of the supersonic into a subsonlo flow (that is, for instance, \diether ateadlly or "by means of a com- pression shock) is doclsively Influenced "by the sonic vclocltj'- "boundary. Of course, the contoiTT of the "body at the "boundary of the Bizpersonic field idiich has "been determined in the foregoing calculation of the example can not coincide with the contour of the original "body idilch was originally distorted according to Frandtl, since the supersonic field, "because of the expansion of the air, requires more space than the flow field ^ich was distorted according to Prandtl. Therefore the surface of the "body in the supersonic region will "be flatter than the surface of the original hody in order to satisfy this Incroeised need for space. The oixter su'bsonlc flow must "be adjusted to this increased need of space of the supersonic field In order to o"bteln -Qie original contour as the result of the calciilatian. This adjustment may "be made "by replacing the supersonic field with dlpoles or source-sin]): distrl"butlons, tdilch will widen tho stroamlines of the su"bsc(Qlc region originally ohtained "by Frandtl* s method. The strength of this source-sink distrl'butlon is Imown fj^Gsn the oc(ndltlcn that the space idilch is to "bo added has to cover tho 8 IfACA. IMNo. 1114 difference of the flov density In eupersonlo flov and flov according to Prandtl and that this space corresponds In magnitude to the flattening of the original "body *hlch resulted in figure 5(a). Bj- a stop-hy-step approximation the contour idilch results at the end of the culculatlon loay thus he adjusted to the desired ocntotcr. Hovever, the eTZonple treated here already permltc the ccncluplon that vbtxa. the local sound \elocity Is exceeded there exist solutions vlth reccnTorslcaa of the local s'aper sonic velocities into suhsonlc velocltloB vlthout conpi'eBslon shock, at least In frlctloolees flow. The pressure dlstrihiitlon >ailch wp.s fo'ind for the example treated.- Is congjared to the prensure 'iJ.sta.'l'bu.ttf-n of tnother slender "body (fig. 5(hJ tixldh was deteminod acc^i'dlnc oo the same lav of confozmal mapping and shculd a^^reo voll td-th the hody of the example. (Soe fig. 5(&).) ^^3 extriiune flitnesa of the pi-esQure dlstrlhutlon In the svpersorlc region in a special chai-acterlBtlc for the hody with local reclon of super- sonic velocity. This hehaviour coiTespmnds to tho sti-ong curvati-re of the contour of the "body imraodlately after the flov has ontered the supersonic region. D-jo to this large lorjal curvature, the hody 3 In this region vill produce even ■for InccsqpresBible flov higher nege.tlve propsurea thpn the hody A. For this reason the two pressure dJ.strlbuticns Tdilch ■♦•ere graphically rrprosentod cpnnot "no 'lireotlj ccoiparod. IV. APEROECMAEE TREMMEHT OP LOC^L EBGIOHS OF SUSBE30KIO VELOCITY The representations of characteristics considered thus far vers notable for the fact that expansion or ccanpresslon vaves running in the same direction never intersect. (Ccsoiparo fig. 5(a).) An Intersecting of vaves running In the same diroctli..n is basically Impossible for expansion vaves since such 'VTUves 'jLLvoys diverge from their starting point. But canrpresalon wives oonvjrge; there- fort^ they could veil foim an envelope and cause a compression shoolc. Such overlapplngs of vaves vere found for the sample body of figure 5(a) '^dien the free-strenm velocity vae inci^eased from M B 0.86 to H a 0.90. The ccnotructlon of the oxpaasian vaves vas started at the sonic boundary (In opposition to their actual direction) and already for the Mach number M a 0.86 (according TIACA TMNo. 232.k to figure ^(a)) led to a pressing together of the expansloti vaves near" the contoizfc'; for* the Mach ninaber M'b 0^90 - they ran Into each other even hefor^ reaching; the oontotir. ^Is condltlan Is physically senseless, that Is, a sjaanetrlcal solution for the sonic houndary iftilch iris found 'according to the Indicated scheme Is, at this Maoh numher M ■ 0^90, no longer posslhle. In this case the "expansion shock,'.' according to a consideration of Frandtl, rather will dissolve Into a group of divergent expansion waves orlg^jiatlng at the contour, vhlle the compression shuck idilch liialts the Erupersonlc area persists. Ill this manner one easily phtalns the flov picture found from many tosts for.'nblch the local sitpersonlc flov Is no longer reconveirted Into the su'bsonlc flov throu^ a contlnfjous phenomenon hut hy means of a cosapresslon shock. The preceding treatment vas hosed upon tho sonic velocity "boundary line idilch vas ohtolned according to Prandbl's rule without additional sou2'ce-3lnk todies wilch woiild have corre- sponded to the groater neei for space of the Buperaaalc reclon. The reeultlng solution tflilch was physically senselesH mi^t veil he caused hy the perhaps no longer appropriate conditions for the exlsteonce of the sonic velocity houndaay line. In order to ohteln a cleeirer vlev one consldei^s the llialtlng case ^ere thT expansion waves converge exactly at the contoirr. A flov around a coxner develops, as represented schemctlcally In flgirre 6; In tho s'.Tpersanlc reclon such a flow Is pooslble vlthout flow separations. Tbin aosunptlon and a given sonic velocity hotmdary offer the supersonic region tho opportizilty to fill the largest posslhle space without concave curvature of the hody In the Bupersonld domain. An admission of concave surfaces would presuppose a sonic velocity houndaiy ■which could no longer he produced from an Incompresslhle flov ahout a profile vlthout additional source and sink dlstrlhutlons In place of the supersonic region; or else one would have to drop the condition of f].ov sjonmetry. ?lgio7e ^(a) demonstrate 8 clearly hov the surface always shows a convex curvatiUTs when the inccanlng wave Is a comprosslon wave, the outgoing wave an expansion wave. However, for the opposite condition of concave curvature In symmetrical flov tlie ccmlng In. of an expansion wave and going out of a compression w&ve would iresult In the amission of expansion waves from the sonic boundary In hoth cases'. But an expansion wave vhlch stairts at the sonic houndary cannot he obtained hy sl3iq;>ly exchanging the cantpresslon end expansion waves, for Instance according to the diagram In figure h; for there must always first appear an expansion wave and thun a compression wave In tho direction of the flov at the sonic velocity houndary, because the flov can 10 KACA OMSo. lull- peisB flxaa the soolo tova&jeaj Into the sttperBonlc region onljr ty means of an expansion, ko. expansion vave stajrtlne kt the sonic botmdazy Is posslMe only for a certain shape of the comers In the graphical representation of characteristics: tiro expansion vaves are sent out from the comer BlTrmltwneov.aljr. (Ccaapeace fig. 7>) There Is, hovever, a condition; the sonic houndaiy In this region must talce a steeper course than the expansion vave vhlch Is Inclined tovard the direction of the flow 'by the Maoh angle. The strecmline through snich a comer point lies hetween the expansion -waves, 'Hhlle for a noimal comer It Is tangent to the apex of the triangle foimed liy the expansion and compresBloa -ware. It la an Important criterion for the comer vLth tvo expansion vsrea that the streamlines heoome steeper, not flatter, vLth Increasing distance from the tody; othei*wlse the flov -Hould not fit together after transition throu^t the two expansion wavas. This reemlt agrees vlth the fact that In a B7mr<jotrlcal supersonic region also the direction of the atreemllnes grows steeper with Increasing distance from the profile. The catise of this phencanenon Is that the stroaaillnes (hecause of the mflTHmnw flov density at Bonlo veloolty) are closest to each other at the sonic boundary vhlle there vlll he the greatest distance hetvoen them for the Btipersonlc region at the point of Tna-rlTmam velocity, that Is, generally, at the jKslnt of greatest thldmess of the hody. Such dletrlhutlons of slopes occur only In supersonic regions; they are not posslhle in suhsonlc areas for a flov ahout hodles vlth convex contotirs. Therefore it la not surprlslnn that the sonic velocity hoxmdary, \dilch vas obtained by distortion of the Incompressible flov about the body, vlll not load to a useful supersonic flov pattern as long as there are no singularities, as for instance source- slnlc distributions in the incompressible flov by means of lAiich the effect of the increased, expansion of the air in the supersonic region can be calculated. Presently a method is under investifjatlon in -vhich th? sonic velocity boundary may be adjusted to tho increasotl need for space of the supersonic flov by arrangement of singularities in the flov; the results of these investigations vlll soon be published separately. Y. SQMSERICAL AFTO msmmsOCAL SBQIORS OF SURERSOHIC FLOV As stated in detail in the preceding chapter, concave body suz^'aces in the supersonic region are dependent t^pon the porticular BACA ON No. lllk 11 type of comers In the net of ohaxaoterlstlcB -Nhlch le characterlEed "by tvo ezpansloD, vavee starting frcoa a point of ^ the soolc 'bottDdazT'. (Canipa3?e fig. 7.) The stlpulatian of flov STmMtry then causes compresslan waves to start from the corresponding points of the contour tdilch In turn cause these cnrnpresslon waves to proceed toward the sonic houndarj In pairs . Since these ccdZQresslon vaves 'vbich Dtaiii at the contour alvays converge, such solutions ere posslhle only as long as the radiated coaiipresslaa vaves do not Intersect; othorvlBe a oonipresBlon shock vlll develop. Therefore the existence of STnanatrlcal flovs vlth local excpersonlc regions cannot "be counted upon for strongly concave curvatinres or for hlgjh Mach numbers, "because the ccsapresslon shoclc -would presuppose a symmetrical expansion shoclc In a corre°spondlng locatloa; li^ls ea^panslon shoclc, hovever, la not physically possllile. ISxcept for the special case descrl'bed aTx>ve, st^ersonlo flows on principle tend toward flow e;mnistry as will he shown In the exanrple represented In figure 8(a). Thore vlll alTrays develop a symmeti-lcal flow alang the wall with two chances In direction for IncQznpressl'ble flow without separations. It Is, however, lmo>qa that for pure supersonic flow the flow along the fTall Is unsynmetrlcaL (fig. 8(a)), If no additional guiding surfaces exist. Conditions of BTnmetry can "be achieved throu^^i calculations for the mixed BuTjoonlc- supersonic flow around this douhle comor: the outer subsonic flow at the sonic houndary will supply the missing synmetrlcal "boundaiy conditions (as for the exEmple In fig, 5(a)). But It does not appear Imposslhle that such a mixed superscnlc- BulsBonlc flow fdilch has "been made symmetrical "by the outer subsonic flov may he unstahle. Since the suhacnlc flov Is produced "by the form of the ohstacle In the' flow It t-jIII "bo symmetrical only for a sjonmetrlcal ohstacle. On the other hand the cuter flow will not "be Bynmietzlcal for an unsymmetrlcal obstacle, as for Instance a symmetrical "body with a local supersonic region and a can^tresslon ehock. Therefor^ It seems very doubtful whether the outer flow vbose fozra la decisively determined "by the foxm of the supersonic region, will In turn "be able to roEihape decisively the supersonic flov by forcing the symmetry conditions upon It.^ A oonf Iznatlon of the noneymnstry of the symnetrloal type of solution would explain the occurrence of unsjianetrlcal solutions with ccoDipresslon shoclcs idilch has been observed In tests. On this, occasion X should lUce to point otct that a ajmnetrlcal supersonic flov also could be suggested for the single wall represented ^A solution of this stability problem shall be obtained by assuming a fgnan unsynnetrlcal defoznatlon of the sonic boundary and by then observing fAiether this defoznatlon Increases or dlmlnlElhes. 12 m.ck m iTo. 1114 in flsare 8(a) idilch Is ^bTsloally not real "but satisfies the conditions for potential flows vltb regard to contlntcLtj and Irrotatlonall ty as veil as Beznoulll's equation. This flotr Is represented jji figure SCb). Such a flow, aa Is vbU Imovn, Is phyBlcally real for the reason that a dlsturljance can not "be tranamltted upstream In supersonic flov. For solutions vhlch vero o1}talned gnalytlcftlly It must, therefore, alvays "be oFipeclally verified ^etber this additional condition has heen satisfied for the supersonic florf. 71. SafMHT 1. A method of calculating the approximate Telocity field for cortpresBlhl© fi-ows vlth local regions of supersTnlc velocity has heon presented. Starting from the flov at a lar^ distance from the profile deteimlned' according to Franltl's rule, this outer flov was continued t-o the sonic Telocity "boundary ty means of a numerical methcd; the metiiod ct characteristics of Frandtl- Busemann was applied for continuation "beyond that "boundary. These calculations result finally In the contour of the profile In the region of supersonic Telocity. 2. It has "been deinonstrated In an exazople that mixed subsonic- supersonic flows a'bcut two-dimensional "bodlos can "be calculated tdiere net only the transition fi-om su'bsonlc to supersonic "but also the transition from. s''rpersonlc to su'bsonlc taJces place continuously, that Is, "vTlthout pressure Jump. 3. ITo physically real local area of supersonic Telocl'ty could "be determined for the example considered here -vihen the sound Telocity was far exceeded "because then esqpanslon or scmpresslon shocks occurred. Howrarer, there Is a prospect of calculating mixed flovB for such cases also: "before starting the calculation one would hare further to extend the oiiter su'bsonlc flow "by means of source and sink distributions; In this way tJie greater need, for sp£u;e of the supersonic Telocity field would "be satisfied. l^. The symmetry of the local area of supersonic Telocity for mixed supersonic- su'bsonlc flows may "be enforced "by the outer su'bsonlc flow. Since, hoverer, the outside subsonic flow In turn must "be produced "by the profile In the flow or "by "the local area of supersonic Telocity, the assuioptlon seems Justified that BAGfL OMNo. 111b 13 often only an Tgista'ble BQnraetrloal flov can "be pcroduced "by the outolde subsonic flov; and this loistable Bjnmetrlcal flov vlll turn even at email dlsturljances Into the unEywmotrloal case vlth cceoopresBlon efhooks. Translated "by M&ry L. Mahler ITatlotoal Mrlsoxy Ccemnlttee for Aoronautlos > Sonic velocity-Boundary 12! o Figure 1. Schematic representation of the flow regions for mixed subsonic -supersonic flow. Fig. 2 NACA TM No. 1114 Figure 2. Partition of the flow field for the difference method. NACA TM No. 1114 Figs. 3,4 ^Q io via.'' Z,6 Figure 3. Relation between air density and velocity under the assumption of an adiabatic change of state. Subsonic re gi on Sonic velocity-Boundary Direction of velocity Supersonic region Figure 4. Sonic boundary with the field of expansion and compression waves, respectively. Sonic velocity-Boundary Expansion waves Compression waves Direction of the oncoming flow (-■• CJI P3 Direction jump in the net of characteristics O > Figure 5a. Grid chacteristics in the local supersonic region for a two-dimensional body B (relation of thickness d/1 = 0.0715 at the Mach number M = 0.86. g i4:>- ipl,, Profile A: Pressu at rfec re distribution i-number M= _Jf^ .-*j — -^ I y ^ ^-'"^ s. ^ N \ 14 IT ^ \ — \ — 1 — \ — r — r 0,2 H/ 1,1 0,1 1,0 o > o >{^ ProfHeAfdll'Umi) Promidli'-om ) Figure 5b. Calculated pressure distribution for two two-dimensional bodies in incompressible flow and in compressible flow with local area of supersonic velocity. Figs. 6,7 NACA TM No. 1114 Expansion waves Sonic velocity-Boundary -^,/x Compression waves Direction of velocity Figure 6. Mixed subsonic-supersonic flow about a body with plane surface in the supersonic region. Normal corners Direction of the oncoming flow Sonic velocity-Boundary Expansion wave Compression wave Figure 7. Schematic representation of the two different kinds of wave radiation at the sonic velocity boundary. NACA TM No. 1114 Fig. 8 Velocity- Vector Compression Figure 8a. Supersonic flow about a double comer. Velocity-Vector Expans i on Expansion ^7777777777 Figure 8b. Symmetrical potential flow with supersonic free-stream velocity about a physically not realizable double corner. NASA Technical Ubrai 3 1 76 01441 2259